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Non-invertibility in Some Heteroscedastic Models

In order to calculate the unobserved volatility in conditional heteroscedastic time series models, the natural recursive approximation is very often used. Following \cite{StraumannMikosch2006}, we will call the model \emph{invertible} if this approximation (based on true parameter vector) converges to the real volatility. Our main results are necessary and sufficient conditions for invertibility. We will show that the stationary GARCH($p$, $q$) model is always invertible, but certain types of models, such as EGARCH of \cite{Nelson1991} and VGARCH of \cite{EngleNg1993} may indeed be non-invertible. Moreover, we will demonstrate it's possible for the pair (true volatility, approximation) to have a non-degenerate stationary distribution. In such cases, the volatility estimate given by the recursive approximation with the true parameter vector is inconsistent.